Energy Considerations in Collisions
Every collision you witness — a cue ball striking the eight ball, a car crash, or even two atoms bouncing off each other — obeys one absolute rule: momentum is conserved. But kinetic energy is a different story. Sometimes kinetic energy is perfectly preserved (like a superball bouncing back to nearly its original height), and sometimes large amounts of it vanish into heat, sound, and deformation (like clay slapping against a wall). Understanding what happens to kinetic energy during a collision is just as important as tracking momentum. Physicists classify collisions into two idealized types based on what happens to kinetic energy. In an elastic collision, the total kinetic energy before and after the collision is exactly the same — no energy is lost to internal processes. In an inelastic collision, some kinetic energy is converted into other forms, so the total kinetic energy after the collision is less than before. A perfectly inelastic collision is the extreme case where the two objects stick together and move as one, producing the maximum possible loss of kinetic energy while still conserving momentum. It is crucial to remember that conservation of momentum always holds in all collisions (when no external forces act), but conservation of kinetic energy only holds in the special case of elastic collisions. Real-world collisions lie somewhere on the spectrum between perfectly elastic and perfectly inelastic. A useful measure of how elastic a collision is is the coefficient of restitution, the ratio of the relative speed after to the relative speed before the collision, ranging from 1 (perfectly elastic) to 0 (perfectly inelastic).
- Identify What Is Always Conserved
- Elastic Collisions: Two Conservation Laws
- Inelastic Collisions: Only Momentum Is Conserved
- Calculating the Kinetic Energy Lost
- Coefficient of Restitution
Identify What Is Always Conserved
In any isolated collision (no net external force), the total momentum of the system is conserved: p_total,before = p_total,after. This is a consequence of Newton's third law — the force object A exerts on B is equal and opposite to the force B exerts on A, so they cancel when you add the momenta of the whole system.
Elastic Collisions: Two Conservation Laws
An elastic collision conserves both momentum AND kinetic energy. Think of ideal billiard balls or atomic/molecular collisions in gases. Mathematically: ½m₁v₁ᵢ² + ½m₂v₂ᵢ² = ½m₁v₁f² + ½m₂v₂f². Combining this with momentum conservation gives exact formulas for the final velocities: v₁f = ((m₁ − m₂)v₁ᵢ + 2m₂v₂ᵢ)/(m₁ + m₂) and v₂f = ((m₂ − m₁)v₂ᵢ + 2m₁v₁ᵢ)/(m₁ + m₂). Notice that if both masses are equal, the first object stops completely and the second moves off with the first object's original velocity — exactly what you see when one billiard ball hits a stationary one straight on.
Inelastic Collisions: Only Momentum Is Conserved
In an inelastic collision, kinetic energy is not conserved — some is transformed into thermal energy, sound, or the energy needed to deform materials. The loss in kinetic energy is ΔKE = KE_before − KE_after, which is always positive (or zero). You can only use momentum conservation to solve for unknowns. For a perfectly inelastic collision where the objects stick together with a common final velocity v_f: m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f. Solving: v_f = (m₁v₁ᵢ + m₂v₂ᵢ)/(m₁ + m₂).
Calculating the Kinetic Energy Lost
Once you find the final velocity of a perfectly inelastic collision, you can calculate how much kinetic energy was converted. Compute KE_before = ½m₁v₁ᵢ² + ½m₂v₂ᵢ² and KE_after = ½(m₁ + m₂)v_f². The energy lost ΔKE = KE_before − KE_after goes into deforming the objects, generating heat and sound. This energy is not 'gone from the universe' — it has just become unavailable as organized kinetic energy.
Coefficient of Restitution
The coefficient of restitution e quantifies elasticity: e = (relative speed of separation)/(relative speed of approach) = |v₂f − v₁f| / |v₁ᵢ − v₂ᵢ|. For a perfectly elastic collision e = 1; for a perfectly inelastic collision e = 0. A tennis ball bouncing on a hard floor has e ≈ 0.75, meaning it loses about 44% of its kinetic energy per bounce (since KE scales as v², and e² gives the fraction of KE retained).
The Key Equation
- \Delta KE — Kinetic energy lost (J): The kinetic energy converted to other forms during the collision; always ≥ 0.
- m_1, m_2 — Masses of objects 1 and 2 (kg): The masses of the two colliding objects.
- \frac{m_1 m_2}{m_1+m_2} — Reduced mass (kg): A single effective mass that captures how both objects share momentum in the centre-of-mass frame.
- v_{1i}-v_{2i} — Relative approach velocity (m/s): How fast the objects approach each other just before collision.
- e — Coefficient of restitution (dimensionless): Ratio of relative separation speed to relative approach speed; 1 = elastic, 0 = perfectly inelastic.
Worked Example
Problem. A 2.0 kg ball moving at 6.0 m/s collides head-on with a stationary 3.0 kg ball. The collision is perfectly inelastic — they stick together. (a) Find the final velocity of the combined mass. (b) Calculate the kinetic energy before and after. (c) How much kinetic energy is lost, and what fraction of the original KE is this?
Answer. Final velocity = 2.4 m/s; KE lost = 21.6 J (60% of the original kinetic energy).
Origins · 1 minJohn Wallis and how this was discovered
John Wallis was one of the first scientists to rigorously formulate the law of conservation of momentum during collisions. In 1668, he presented a paper to the Royal Society analyzing the collision of imperfectly elastic bodies, correctly identifying that the total momentum (which he called 'quantity of motion') is conserved even when kinetic energy is not — a conceptual breakthrough that separated the two conservation laws for the first time. Wallis was also the mathematician who introduced the symbol ∞ for infinity, still used worldwide today. He was largely self-taught in mathematics and learned it as a hobby during the English Civil War.
Related Concepts
- Conservation of Momentum — Momentum conservation is the foundational law that applies to all collisions, elastic or inelastic, and must be applied before any energy analysis.
- Impulse and Momentum — Impulse describes the force-time interaction during a collision that changes momentum, providing a microscopic view of how collisions work.
- Work-Energy Theorem — Understanding how work done by internal forces during a collision converts kinetic energy into other forms requires the work-energy theorem.
- Conservation of Energy — Total energy (including thermal and sound) is always conserved; only kinetic energy can decrease, with the remainder transforming into other energy forms.